Respuesta :
The formula for a z-score is
[tex]z=\frac{X- \mu}{\sigma}[/tex]
We don't have μ, the mean, or σ, the standard deviation. We can, however, set up a system of two equations with the information we do have. We know:
[tex]-3=\frac{60- \mu}{\sigma} \text{ and } 2=\frac{85- \mu}{\sigma}[/tex]
We can make this look more like a "typical" system of equations by cancelling the fractional part of each equation. We do this by multiplying the bottom of each equation by σ:
[tex]-3*\sigma=\frac{60- \mu}{\sigma}*\sigma \text{ and } 2*\sigma=\frac{85- \mu}{\sigma}*\sigma \\ \\-3\sigma=60- \mu \text{ and } 2\sigma=85- \mu[/tex]
Now we'll get the variables on the same side of the equation. We'll do this by adding μ to both sides:
[tex]-3\sigma + \mu=60- \mu + \mu \text{ and } 2\sigma+ \mu =85- \mu + \mu \\ \\-3\sigma+\mu=60 \text{ and } 2\sigma+\mu=85[/tex]
Now we'll write this vertically:
[tex] \left \{ {{-3\sigma+\mu=60} \atop {2\sigma+\mu=85}} \right. [/tex]
We will "eliminate" μ by subtracting the bottom equation:
[tex] \left \{ {{-3\sigma+\mu=60} \atop {-(2\sigma+\mu=85)}} \right. \\ \\-5\sigma=-25[/tex]
Divide both sides by -5:
[tex]\frac{-5\sigma}{-5}=\frac{-25}{-5} \\ \\\sigma=5[/tex]
Our standard deviation is 5. Substituting this into our top equation we have:
-3(5)+μ=60
-15+μ=60
Cancel the -15 by adding it to both sides:
-15+μ+15=60+15
μ=75
Our mean is 75.
[tex]z=\frac{X- \mu}{\sigma}[/tex]
We don't have μ, the mean, or σ, the standard deviation. We can, however, set up a system of two equations with the information we do have. We know:
[tex]-3=\frac{60- \mu}{\sigma} \text{ and } 2=\frac{85- \mu}{\sigma}[/tex]
We can make this look more like a "typical" system of equations by cancelling the fractional part of each equation. We do this by multiplying the bottom of each equation by σ:
[tex]-3*\sigma=\frac{60- \mu}{\sigma}*\sigma \text{ and } 2*\sigma=\frac{85- \mu}{\sigma}*\sigma \\ \\-3\sigma=60- \mu \text{ and } 2\sigma=85- \mu[/tex]
Now we'll get the variables on the same side of the equation. We'll do this by adding μ to both sides:
[tex]-3\sigma + \mu=60- \mu + \mu \text{ and } 2\sigma+ \mu =85- \mu + \mu \\ \\-3\sigma+\mu=60 \text{ and } 2\sigma+\mu=85[/tex]
Now we'll write this vertically:
[tex] \left \{ {{-3\sigma+\mu=60} \atop {2\sigma+\mu=85}} \right. [/tex]
We will "eliminate" μ by subtracting the bottom equation:
[tex] \left \{ {{-3\sigma+\mu=60} \atop {-(2\sigma+\mu=85)}} \right. \\ \\-5\sigma=-25[/tex]
Divide both sides by -5:
[tex]\frac{-5\sigma}{-5}=\frac{-25}{-5} \\ \\\sigma=5[/tex]
Our standard deviation is 5. Substituting this into our top equation we have:
-3(5)+μ=60
-15+μ=60
Cancel the -15 by adding it to both sides:
-15+μ+15=60+15
μ=75
Our mean is 75.
